True mastery requires practicing the habits and authentic performances of a discipline, not simply the skills.
A recent research summary about an idea called “overlearning” intrigued me (you can read the article here). Even though the research wasn’t specific to elementary classrooms, there were some ideas included that are worth considering when it comes to deciding how much math practice a student needs. Is there such a thing as “too much practice” or “overlearning” in math? Should there be something different about elementary math practice than exists today?
As a math teacher, I’ve spoken with many parents and educators who expect students to have plenty of math practice—but not TOO much. Of course no one wants a student to practice something when they already “get it,” but what do we mean by “get it”; what is there to “get” and what is “it”?
Sometimes I wonder if people view math as something that needs to be practiced in a unique way—somehow different from other subjects or activities. For example, when a child is becoming fluent in spoken language, there’s no such thing as “too much practice” speaking in that language. Parents don’t say, “My three-year-old is having WAY TOO MANY conversations each day.” Some young children do talk a lot, but at least in a learning context it’s not considered a bad thing. In fact, it’s a very good thing for becoming fluent because children are not simply “practicing” the language, they are engaging in the authentic performance of communication and social interaction. We don’t tell children, “Go practice telling people about what you had for breakfast. Tell at least four people.”
Here are two related quotes from the “overlearning” article that stood out to me:
“The perfect execution of … a tennis serve doesn’t mark the end of practice; it signals that the crucial part of the session is just getting underway.”
“We have shown there is an advantage to continued practice beyond any visible changes in performance.”
While conceptual understanding and skill in math is not completely analogous to serving a tennis ball, it’s important to note that what we’re considering is the evidence of “visible changes in [a learner’s] performance.” It’s also important to note that a serve is only one part of the overall performance of a tennis match. Just as a player could have a great serve but perform a terrible overall game of tennis, it’s possible for students to demonstrate a math skill perfectly, but lack number sense and fluency.
What exactly needs to be practiced to develop number sense and fluency? For example, if a student decides he absolutely must use the long division algorithm to determine that 1,001 divided by 25 will have a remainder of 1, then it’s likely his mental math skills need improvement. But too often, a student’s number sense isn’t assessed using problems such as this one, and therefore it also isn’t developed or practiced in this way. Students are frequently assigned to practice the long division algorithm rather than to engage in an exercise that deepens their number sense, mental fluency, and ability to know whether an answer is reasonable. All are important learning outcomes, but not all of them are allocated the same amount of practice time. The performance of a tennis player includes much more than just the serve, and the performance of a mathematician includes far more than simply computation. For that reason, whenever we ask, “HOW MUCH practice is needed?” we first need to ask, “WHAT needs to be practiced?”
Are your students having trouble solving problems in new situations? They need to regularly encounter unfamiliar problems in their practice sessions.
People often think that math study mostly involves skill practice. But when teachers are asked, “In what areas do students struggle the most?” the answer is commonly, “Problem solving in new situations.” Therefore, it would seem that students need more practice thinking mathematically in new situations than they currently experience. It’s tough to require students to regularly practice solving unfamiliar, non-routine problems because they’ll frequently say, “But you haven’t shown me how to do it.” But it’s absolutely necessary.
How do The Standards for Mathematical Practice lead to real conceptual understanding?
In this sense, students in states that have adopted the Common Core State Standards will benefit from “Practicing the Practices”—that is, spending more time engaging in the eight Standards for Mathematical Practice.
If students spend more time practicing the mathematical performances of “constructing arguments,” “critiquing arguments,” and “looking for structure,” then their mathematical thinking will improve significantly, along with their conceptual understanding of important skills. I’m not sure I’d call that “overlearning,” but I do believe that it would ensure long-term understanding and transfer in the minds of students as well as greater math mastery. Students need to practice more than simply skills.
How does DreamBox support practice and math mastery?
When students use DreamBox, they are practicing skills while they are simultaneously developing conceptual understanding and fluency using the virtual manipulatives. Our lessons continually engage students in the practices of mathematicians (including the CCSS Practices), so they routinely practice those critical thinking habits as well. When students complete a unit or achieve a proficiency in DreamBox, it is because they have experienced enough practice to have demonstrated strong evidence of their mathematical understanding and performance.
“Overlearning?” Too much math practice? It all depends on what students are practicing.